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Physics LibreTexts

1.P Exercises

  1. According to classical physics, a non-relativistic electron whose instantaneous acceleration is of magnitude $ a$ radiates electromagnetic energy at the rate

     

    $\displaystyle P = \frac{e^2\,a^2}{6\pi\,\epsilon_0\,c^3},
$

     

    where $ e$ is the magnitude of the electron charge, $ \epsilon_0$ the permittivity of the vacuum, and $ m_e$ the electron mass. Consider a classical electron in a circular orbit of radius $ r$ around a proton. Demonstrate that the radiated energy would cause the orbital radius to decrease in time according to

     

    $\displaystyle \frac{d}{dt}\left(\frac{r}{a_0}\right)^3 =- \frac{1}{\tau},
$

     

    where $ a_0=4\pi\,\epsilon_0\,\hbar^2/(m_e\,e^2)$ is the Bohr radius, $ \hbar$ the reduced Planck constant, and

     

    $\displaystyle \tau=\frac{a_0}{4\,\alpha^4\,c}.
$

     

    Here, $ c$ is the velocity of light in a vacuum, and $ \alpha=e^2/(4\pi\,\epsilon_0\,\hbar\,c)$ the fine structure constant. Deduce that the classical lifetime of a hydrogen atom is $ \tau\simeq 1.6\times 10^{-11}\,{\rm s}$ .

     

  2. Demonstrate that

     

    $\displaystyle \langle B\vert A \rangle = \langle A\vert B \rangle^\ast
$

     

    in a finite dimensional ket space.

     

  3. Demonstrate that in a finite dimensional ket space: Here, $ X$ , $ Y$ are general operators.

     

    1.  

      $\displaystyle \langle B\vert\,X^{\dag }\,\vert A\rangle = \langle A\vert\,X\, \vert B\rangle^\ast.
$

       

    2.  

      $\displaystyle (X\, Y)^{\dag } = Y^{\dag }\, X^{\dag }.$

       

    3.  

      $\displaystyle (X^\dag )^\dag = X.
$

       

    4.  

      $\displaystyle (\vert B\rangle\langle A\vert)^\dag = \vert A\rangle\langle B\vert.
$

       

  4. If $ A$ , $ B$ are Hermitian operators then demonstrate that $ A\,B$ is only Hermitian provided $ A$ and $ B$ commute. In addition, show that $ (A+B)^n$ is Hermitian, where $ n$ is a positive integer.

     

  5. Let $ A$ be a general operator. Show that $ A+A^\dag$ , $ {\rm i}\,(A-A^\dag )$ , and $ A\,A^\dag$ are Hermitian operators.

     

  6. Let $ H$ be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator $ \exp(\,{\rm i}\,H)\equiv\sum_{n=0,\infty}
({\rm i}\,H)^n/n!$ is $ \exp(-{\rm i}\,H)$ .

     

  7. Let the $ \vert\xi'\rangle$ be the eigenkets of an observable $ \xi$ , whose corresponding eigenvalues, $ \xi'$ , are discrete. Demonstrate that

     

    $\displaystyle \sum_{\xi'} \vert\xi'\rangle \langle \xi'\vert = 1,$

     

    where the sum is over all eigenvalues, and $ 1$ denotes the unity operator.

     

  8. Let the $ \vert\xi_i'\rangle$ , where $ i=1,N$ , and $ N>1$ , be a set of degenerate eigenkets of some observable $ \xi$ . Suppose that the $ \vert\xi_i'\rangle$ are not mutually orthogonal. Demonstrate that a set of mutually orthogonal (but unnormalized) degenerate eigenkets, $ \vert\xi''_i\rangle$ , for $ i=1,N$ , can be constructed as follows:

     

    $\displaystyle \vert\xi''_i\rangle = \vert\xi'_i\rangle - \sum_{j=1,i-1}\frac{\l...
...'\vert\xi_i'\rangle}{\langle \xi_j''\vert\xi_j''\rangle}\,\vert\xi_j''\rangle.
$

     

    This process is known as Gram-Schmidt orthogonalization.

     

  9. Demonstrate that the expectation value of a Hermitian operator is a real number. Show that the expectation value of an anti-hermitian operator is an imaginary number.

     

  10. Let $ H$ be an Hermitian operator. Demonstrate that $ \langle H^{\,2}\rangle \geq 0$ .

     

  11. Consider an Hermitian operator, $ H$ , that has the property that $ H^{\,4}=1$ , where $ 1$ is the unity operator. What are the eigenvalues of $ H$ ? What are the eigenvalues if $ H$ is not restricted to being Hermitian?

     

  12. Let $ \xi$ be an observable whose eigenvalues, $ \xi'$ , lie in a continuous range. Let the $ \vert\xi'\rangle$ , where

     

    $\displaystyle \langle \xi'\vert\xi''\rangle = \delta(\xi'-\xi''),$

     

    be the corresponding eigenkets. Demonstrate that

     

    $\displaystyle \int d\xi'\, \vert\xi'\rangle\langle \xi'\vert= 1,
$
    where the integral is over the whole range of eigenvalues, and $ 1$ denotes the unity operator.

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